MATH 175: Numerical Analysis II Lecturer: Jomar F. Rabajante IMSP, UPLB 2nd Sem AY 2012-2013

ACCELERATING CONVERGENCE: Aitken’s ∆2 process Used to accelerate linearly convergent sequences, regardless of the method used. Actually, this acceleration method is not only applicable to root- finding algorithms.

ACCELERATING CONVERGENCE: Aitken’s ∆2 process F O R M U L A (update the value of rk)

ACCELERATING CONVERGENCE: Aitken’s ∆2 process Actually, Aitken’s process is an extrapolation. r delta r delta^2 r rk-2 rk-1 rk-1-rk-2 rk rk-rk-1 (rk-rk-1)-(rk-1-rk-2)

ACCELERATING CONVERGENCE: Aitken’s ∆2 process Steffensen’s Method: a modified Aitken’s delta-squared process applied to fixed point iteration For sample computations, see the MS Excel file.

Other Method: Muller’s Method Extension of secant method – instead of using linear interpolation, it uses quadratic interpolation (parabola) May generate complex zeros (use software that can understand complex arithmetic) Less sensitive to starting values compared to Newton’s Method Order of convergence: p≈1.84

Other Method: Muller’s Method Initial points: f parabola

Other Method: Muller’s Method Formula z1,z2 & z3 came from Newton’s Divided Difference, and xk came from quadratic formula